On a new property of $n$-poised and $GC_n$ sets

TL;DR

This paper investigates properties of $n$-poised and $GC_n$-sets, proving that under the Gasca-Maeztu conjecture, $n$-node lines are used by a specific number of nodes, revealing new phenomena in these sets.

Contribution

It establishes new results about the usage of $n$-node lines in $n$-poised and $GC_n$-sets, assuming the Gasca-Maeztu conjecture, and introduces phenomena not previously observed.

Findings

If the Gasca-Maeztu conjecture holds, $n$-node lines are used by exactly $inom{n}{2}$ or $inom{n-1}{2}$ nodes.

Similar usage patterns are shown for $n$-node or $(n-1)$-node lines in general $n$-poised sets.

The paper proposes a conjecture about the usage of any $k$-node line.

Abstract

In this paper we consider n-poised planar node sets, as well as more special ones, called $GC_n$-sets. For these sets all $n$-fundamental polynomials are products of n linear factors as it always takes place in the univariate case. A line ${\ell}$ is called $k$-node line for a node set $\mathcal X$ if it passes through exactly $k$ nodes. An $(n+1)$-node line is called maximal line. In 1982 M. Gasca and J. I. Maeztu conjectured that every $GC_n$-set possesses necessarily a maximal line. Till now the conjecture is confirmed to be true for $n \le 5$. It is well-known that any maximal line $M$ of $\mathcal X$ is used by each node in $\mathcal X\setminus M,$ meaning that it is a factor of the fundamental polynomial of each node. In this paper we prove, in particular, that if the Gasca-Maeztu conjecture is true then any $n$-node line of $GC_n$-set $\mathcal X$ is used either by exactly $\binom{n}{2}$ nodes or by exactly $\binom{n-1}{2}$ nodes. We prove also similar statements concerning $n$-node or $(n-1)$-node lines in more general $n$-poised sets. This is a new phenomenon in $n$-poised and $GC_n$ sets. At the end we present a conjecture concerning any $k$-node line.